PRELIMINARY web page. By Warren D. Smith, Nov. 2009.
A single-winner voting system "fails the NESD property" if, when every honest voter changes her ballot to now rank A top and B bottom (or B top and A bottom; which depends on the voter), leaving it otherwise unaltered, that always (except in very rare "exact tie" situations) causes A or B to win.
Presumably voting systems failing NESD will tend to cause a country to become 2-party dominated.
We also define the "NESD*" property to be the same as NESD except A and B are to be solely-top-rated or ranked by all voters; we forbid coequal top.
Systems failing NESD: IRV (instant runoff voting), plurality, and all Condorcet systems. [By these I mean, with pure-rank-order ballots – no rank-equalities permitted.]
Systems passing NESD: Borda, approval and range voting.
However: Borda restricted to 3-candidate elections fails NESD.
If we modify IRV to permit rank equalities by counting a ballot with K candidates co-equal top as 1/K votes for each, then this system passes NESD, although I still feel uncomfortable about it because being co-equal top is plainly a lot worse and more vulnerable than being sole-top (there is kind of a "discontinuity," unlike in range voting where it is "continuous" as you move across the top score), so strategic voters might not do the former.
This is made clearer via an analogy. Consider plurality voting with "equal votes" permitted (i.e. you can vote "half" for Jefferson and "half" for Adams). This would clearly be stupid, i.e. would clearly be essentially equivalent strategically to plain plurality. Why? Consider each of your half-votes one at a time. If for the first, your best move was to vote Jefferson; then for the second, the same reasoning should apply. Hence you'd vote 100% for Jefferson and never use the equality feature (unless you were a strategic idiot). Plurality with equals permitted, does technically pass NESD, but this reasoning suggests that is a misleading perception. Really, to avoid being misled, one instead should consider the NESD* property.
If we consider Condorcet systems with rank-equalities permitted, these also technically pass NESD but again there is that worrying "discontinuity." So we now enquire which voting systems pass NESD*:
Fail NESD*: IRV, plurality, Condorcet (all with rank-equalities permitted or forbidden, both ways they fail NESD*).
Pass NESD*: Range voting.
NESD* not applicable: Approval voting.
NESD stands for "Naive Exaggeration Strategy ⇒ Duopoly."
"NES" refers to the voter strategy of
The "D" part means: if all (or a very large percentage) of voters exhibit NES behavior, then one of the apparent-top-two candidates will always win (except in exceedingly unlikely "perfect-tie" scenarios).
And in fact, the same winner will arise as in strategic plurality voting, so any system failing NESD or NESD* can be accused (perhaps not with full justification, but certainly with some) of being "equivalent in the real world" to plain plurality voting. If so, it presumably over historical time will yield "duopoly," aka "2-party domination" where voters effectively only get one of two choices (or no choice) every election. This severely diminishes voter choice and "democracy" (as opposed to some system with more than 2 choices).
It's an interesting property (or two properties if we count NESD* also) and I think worth consideration.
You can now ask yourself other interesting questions, like "how can I design good voting systems passing NESD or NESD*?" etc.
Definition: A voting system obeys "Majority-Top" if, whenever a voter-majority ranks X unique-top in their vote, then X must win the election.
Jonathan Lundell pointed out the (obvious, once you see it) fact that the (apparently desirable) NESD* property, conflicts with the (also apparently desirable) Majority-Top property: It is impossible for a voting system to obey both.
My personal lesson from this: Majority-Top is not an always-desirable property for voting systems, despite the common perception that it "obviously" is. Indeed, consider a range voting (0 to 9) election like this:
| #voters | Their Vote |
|---|---|
| 40 | A=9, B=0, C=7 |
| 51 | A=0, B=9, C=7 |
| 9 | A=0, B=0, C=9 |
A 51% voter-majority scores B unique-top. Therefore, in any voting system obeying Majority-Top, B must win. However, it seems pretty clear in this situation that C would generally be the best winner for society. And it is the range voting winner (totals: C=718, B=459, A=360) and also the one who would probably be elected by approval voting.
This situation often arises where there are two power-groups, the A-ites and the B-ites, each of whom wishes to destroy the other. If the A-ites gain a slight majority, game over. However, it could be there is some middle course ("C") which everybody, regardless of their power-group, finds reasonably attractive, although comparatively few regard C as the best option for them. Should it really be demanded, as a core principle of voting systems, that this middle course must, under all circumstances, always lose? I think not.
If you agree, even in a single election instance, then you must drop the Majority-Top property as a demand. (Remember, this logical property brooks no exceptions. You, by agreeing with even a single exception, disagree with the logical property.)
An analogous example-election in the rank-order-ballot world might be
Do you agree with me that declaring C, not B, winner would usually be better for society?
#voters Their Vote 40 A>C>P>Q>R>S>B 51 B>C>S>R>Q>P>A 9 C